Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  

Series: Mechanical Engineering Vol. 12, N
o
 1, 2014, pp. 73 - 84 

DESIGN OF SMALL BULB TURBINES WITH UNEQUAL 

SPECIFIC WORK DISTRIBUTION OF THE RUNNER'S 

ELEMENTARY STAGES


UDC 621.22 

Jasmina Bogdanović-Jovanović
1
, Božidar Bogdanović

1
, Ivan Božić

2

1
University of Niš, Faculty of Mechanical Engineering 

 2
University of Belgrade, Faculty of Mechanical Engineering 

Abstract. Regarding the present state of knowledge in the field of the turbomachinery 

design, the method for designing small bulb turbines with unequal specific work 

distribution of the turbine runner's elementary stages near the hub is presented in the 

paper. The distribution function of specific work of all the elementary stages is 

obtained, according to which the averaged axisymmetric flow surfaces of the turbine 

runner have a negligibly small deviation from the cylindrical flow surfaces. The 

specific work of the near-the-hub elementary stages, in the given distribution function, 

can be reduced up to 60% of the required (design) specific work, still achieving nearly 

cylindrical flow surfaces.     

Key Words: Bulb Turbine, Elementary Stages, Specific Work, Axisymmetric Flow 

Surfaces 

1. INTRODUCTION

Both the hydraulic turbines design and their operating performance analysis require 

the use of simpler methodologies in preliminary design phases, especially when the 

geometry of the turbine runner is not completely defined. In the field of the 

turbomachinery design, such attempts are continuously made thus providing us with 

different methodologies and procedures for the design process optimization [10, 11]. 

With the computer technology development, a significant breakthrough of numerical 

methods and numerical simulation of the fluid flow has been made, encouraging the 

numerical techniques incorporation into the design procedure and performance analysis of 

the turbomachinery [12, 13].  

Received November 15, 2013 / Accepted January 19, 2014


Corresponding author: Jasmina Bogdanović- Jovanović 

University of Niš, Faculty of Mechanical Engineering, Department of Hydroenergetics, Niš, Serbia 

E-mail: bminja@masfak.ni.ac.rs 

Original scientific paper



74 J. BOGDANOVIĆ-JOVANOVIĆ, B. BOGDANOVIĆ, I. BOŽIĆ 

A number of assumptions and simplifications, empirical equations as well as the 

designer's experience always accompany the designing process [1, 2, 5, 7, 9]. The basic 

assumption in the hydraulic turbine designing procedure is axisymmetric flow surfaces. An 

elementary stage is a flow space between two elementary immediate axisymmetric flow 

surfaces. The intersection of stay vane (SV), or turbine runner (TR), and axisymmetric flow 

surfaces, defines the vane or blade profiles in the turbine elementary stages. 

Stay vanes and blades are formed by the stay vane profiles and the runner blade 

profiles in the turbine's elementary stages. The vane and blade profiles in the turbine 

elementary stages can be defined using suitable hydraulic calculations for a two-

dimensional fluid flow through the profile cascade of axisymmetric flow surfaces. In the 

cases when the flow surfaces are cylindrical, the theory of fluid flow through the straight 

plane profile cascade can be used in calculations [6]. 

In order to achieve cylindrical or almost cylindrical flow surfaces in the bulb turbine 

runners, these runners are usually designed to obtain an equal specific work of all the 

elementary stages. The turbine runners designed according to such a principle are made of 

blade profiles with a significantly larger inclination angle near the hub than near the shroud. 

In order to minimize the blades' spatial curvature and to reduce the axial length of the 

runner, it is eligible to design the runner with smaller specific work of the elementary 

stage nearer the hub than the blade periphery [3, 4]. In this way, the spatial curvature of 

the stay vanes is also reduced. The possibility to design such stay vanes and turbine 

runners is analyzed in the paper. Only small bulb turbines, which can be used in small 

hydropower plants or as models of large water turbines (for the model testing purpose) 

are considered in this study.   

2. BASIC FORMULAS 

The scheme of a bulb turbine meridional cross-section, with traces of two elementary 

immediate axisymmetric flow surfaces, where one (below) is marked as Sm, is presented 

in Fig. 1. A flow space between two elementary immediate axisymmetric flow surfaces 

represents an elementary stage of the runner. To determine the shape of stay vanes and 

runner blades, it is sufficient (for small turbines) to define profiles in 7 to 12 elementary 

stages, approximately evenly distributed along the blade height. 

 

Fig. 1 Scheme of a bulb turbine meridional cross-section  



 Design of Small Bulb Turbines with Unequal Specific Work Distribution of the Runner's Elementary Stages  75 

The runner is designed as a free-vortex flow (to obtain zero circumferential 

component of absolute flow velocity at outlet (cu2=0)); therefore, based on the Euler's 

equation, the specific work of the turbine runner elementary stage is: 

 
1 1

( ) ( ) ( )
k m m u m

y S r S c S   ,   za 
2
( ) 0

u m
c S  , (1) 

where: cu, cr, cz – circumferential, radial and axial component of absolute flow velocity 

[m/s] (absolute flow velocity 
o o o

c= u r z
u r z

c c c  ), r1 – radius in cross-section 1 [m],  – 

angular velocity of turbine runner [s
-1

]. 

 According to Eq. (1), the required circumferential component of absolute flow 

velocity, in front of the inlet of the turbine elementary stage, can be defined by formula:  

 
1

1

( )
( )

( )

k m
u m

m

y S
c S

r S



. (2) 

Stay vanes produce circumferential components of absolute velocity in front of the 

runner inlet. Neglecting the influence of viscous friction, the flow in area between SV and 

TR (vaneless area) is free-vortex flow: 

 
1 1' 1 1

( ) ( ) . ( ) ( ) ( ) ( )
m u m m u m m u m

r S c S const r S c S r S c S      ,  

therefore, the circumferential components of absolute velocity of the stay vane elementary 

stages can be calculated as:  

 
1

1' 1

1 1

( ) ( )
( ) ( )

( ) ( )

m k m
u m u m

m m

r S y S
c S c S

r S r S 
 


. (3) 

Streamline inclination angle (with respect to circumferential coordinate 
o

ur  , where 
o

u  is the unit vector in the direction of circumferential velocity u , 
o

u = uu  ,  = u r  ) 
in the outlet of stay vane area is calculated by the next formula:  

 
1'

1

1'

( )
( )

( )

m m
m

u m

c S
S arctg

c S
  ,    where 

2 2
=

m r z
c c c . (4) 

To determine the inclination angles of vane profile chamber line, in the outlet of the 

stay vane elementary stages,
l. l .1

( ) ( ) ( )
b m m b m

S S S     , it should be assumed that the 

value of flow inclination angle in outlet of the stay vane is (
o

.1
( ) (2 5)

b m
S   ).  

In front of the inlet of the stay vane, there is a zero circumferential component of the 

absolute velocity (cu0 = 0), thus, the inclination angle of the camber line in the inlet of the 

turbine stay vane is 1.a(Sm) = 90
o
. 

Design operating parameters of the turbine are: , Q = Q
+
, YT = YT

+
, where Q

+
 and YT

+
 

are the best efficiency (optimal) operating parameters (when the turbine operates with 

maximum efficiency,  = max =  
+
), Q [m

3
/s] is turbine volume flow rate and YT [J/kg] 

is turbine specific work (
T T

Y gH  (g = 9.81 m/s
2
), HT  – net turbine head [m]). 

 Adopting the maximum value of hydraulic efficiency (h = h
+
), design specific work 

of the turbine runner is Yk = Yk
+ 

= h
+ 

 YT
+
. When designing a turbine using the model of 

equal specific work of turbine elementary stages (yk (Sm) = const. = Yk
+
), and when flow 

surfaces in the runner are cylindrical (r1(Sm) = r2(Sm), cz1 = cz2 = const.), the theory of flow 



76 J. BOGDANOVIĆ-JOVANOVIĆ, B. BOGDANOVIĆ, I. BOŽIĆ 

in the straight plane profile cascades is used. Inclination angles of profile chord in 

elementary stages of the turbine runner (t = t (Sm)) are defined by the lift force method, 

where, using the above mentioned theory, the well known formula is used [1, 2]: 

 =2
u

y

wl

t w





,     (5) 

where: y – profile lift coefficient, l – length of the chord line, t – cascade spacing,  

wu = wu1  wu2 (wu = cu, for cu2 = 0), w = c u  is relative velocity of water flow, 
o o o

w= u r z
u r z

w w w  ( wu = cu  u, wr = cr , wz = cz) and w – value of averaged velocity 

in infinity (
1 2

w 0.5(w w )

  ). 

For elementary stages of the runner there are: y = y (Sm), t  = t (Sm), l  = l (Sm), 

wu = wu (Sm) and w = w (Sm). Since coefficient y  depends on inclination angle of the 

profile chord line, based on formula (5), for calculated valuey , the inclination angle of 
the profile chord line can be determined respectively.  

In the case where flow surfaces in the runner are not cylindrical (when flow 

components are not uniform along the radius), volume flow rate Q (Sm) passing through 

the flow space between the hub and flow surface Sm, is calculated as follows: 

 
1 2( ) ( )

1 2
( ) 2 ( ) 2 ( )

m m

i i

r S r S

m z z

r r

Q S c r rdr c r rdr     , (6) 

where, cz1(r) and cz2(r) are distribution functions of axial components of absolute flow 

velocities in the control cross-sections in front of and behind the runner. Functions cz1(r)  

and cz2(r) should satisfy the following formula of volume flow through the runner:   

 
1 2

2 ( ) 2 ( )
e e

i i

r r

z z

r r

Q c r rdr c r rdr     . (7) 

For given function yk (r), the specific work of turbine runner can be calculated using 

the formula: 

 

2

2

1 1
( ) 2 ( ) ( )

e

i

r

k k k z

A r

Y y r dQ y r c r rdr
Q Q

 
    
 
 

  . (8) 

3. CONDITIONS FOR OBTAINING CYLINDRICAL FLOW SURFACES IN THE CONTROL CROSS-

SECTIONS IN FRONT OF AND BEHIND THE TURBINE RUNNER 

The necessary condition is that the control cross-sections are placed in the space 

which is physically bounded by cylindrical surface of the hub (turbine hub radius 

ri = const.) and the shroud (turbine shroud radius re = const.), as shown in Fig. 1.  

The equation of steady fluid flow of the inviscid fluid, disregarding gravity acceleration, 

can be written in the form:  



 Design of Small Bulb Turbines with Unequal Specific Work Distribution of the Runner's Elementary Stages  77 

  
1

,
t

c rotc gradp


  , (9) 

where p is static pressure [Pa] and pt  the total pressure (
2

/ 2
t

p p c  ) [Pa]. 

For the fluid flow on cylindrical surfaces (cr = 0, 
o o

c u z
u z

c c  ) in control cross-sections 

in front of and behind the turbine runner (where / 0   ), according to the component of 

vector Eq. (9) in the direction of coordinate r, a differential equation is obtained: 

 
( ) 1u u tz

z

c rc pc
c

r r r r

 
 

   
, (10) 

which defines the condition for cylindrical flow surfaces in control cross-sections 1-1 and 2-2. 

In control cross-section 2-2 (outlet of the runner) cu2 = 0 and pt = pt,2 = const., 

therefore, according to equation (10) it is cz = cz2 = const.  

Defining the hydraulic efficiency of the runner elementary stage as: 

 

1 2 1 2

.

( ) ( )
( )=

( ) ( ) ( )

k m k m
h k m

t m t m t m

y S y S
S

p S p S p S

 




 


 
. (11) 

and, assuming that hydraulic efficiency is the same for all elementary stages 

(
. .

( ) = .
h k m h k

S const  ), the total pressure in control cross-section 1-1 (inlet of the 

runner) can be expressed as: 

 
1 2

1

.

( )
( )

k
t t

h k

y r
p r p




  . (12) 

According to Eq. (1): 

 
1 1

1
( )

u k
rc y r


 ,    i.e.  

1 1

1
( )

u k
c y r

r
  (13) 

where 
1

( ) ( )
k k m

y r y S , due to 
1 1

( )
m

r r S . 

Considering Eqs. (12) and (13), Eq. (10) for obtaining cylindrical surfaces in the 

control cross-sections 1-1 (for 
1 1
( )

t t
p r p , 

1z z
c c , 

2u u
c c ) yields: 

 11
1 12 2

.

( )1 1
( )

kz
z k

h k

y rc
c y r

r rr

  
  

   
. (14) 

where: 
2 2

.

( )1
0

k

h k

y r

r
 

 
. 

For equal specific work of the runner elementary stages ( ( )k ky r const Y


  ), due to 

equation (14) it is obtained cz1 = const., and due to Eqs. (6) and (7) cz1 = cz2 and r1(Sm) = 

r2(Sm), thus the assumption of the cylindrical flow surfaces in the runner is reasonable [8]. 

If  / 0
k

y r   , according to equation (14), 
1

/ 0
z

c r   , thus, due to equation (5), for 

2 z
const. c

z
c   , can be resolved that 

1 2
( ) ( )

m m
r S r S  and flow surfaces are approximately 

conical in the runner. 



78 J. BOGDANOVIĆ-JOVANOVIĆ, B. BOGDANOVIĆ, I. BOŽIĆ 

Further on, the relationship between the hydraulic efficiency of turbine runner 

(
.

( )
h k m

S ) and turbine hydraulic efficiency ( ( )
h m

S ) is obtained. 

Denoting 
.t I

p  and 
.t II

p  as total pressures on inlet (I) and outlet (II) of turbine, the 

hydraulic efficiency of the turbine elementary stage is defined by relation: 

  
. .

( ) ( )

( ) ( ) ( )

k m k m
h m

T m t I m t II m

y S y S
S

y S p S p S


  


. (15) 

Since 
1 1 2 2 1 1 2 2. .

( ) ( )
I II I IIt I m t II m t t t g g g

p S p S p p p y y y  
     

        , where 
1Ig

y


 

and 2 IIgy   are specific losses of flow energy in the flow passages from I to 1 and the flow 

passages from 2 to II, it is easy to predict that there is a relationship between ( )
h m

S  and 

.
( )

h k m
S : 

 
1

. 1 2

1

( )
h

h k I II



 

 
   

,   or  
. 1

1 2

1

( )
h k

h I II



 

 
   

, (16) 

where, ( )
h h m

S  , 
. .

( )
h k h k m

S  , 
1 1

( )
I I m

S 
 
  and 

2 2
( )

II II m
S 

 
  are dimensionless 

losses of flow energy in the passages from I to 1 and the flow passages from 2 to II 

(turbine diffuser),  

 
. 1

1 1

( )
( )

( )

g I m

I I m

k m

y S
S

y S



 
       and  

.2

2 2

( )
( )

( )

g II m

II II m

k m

y S
S

y S



 
    . (16') 

Since 
1 2 .1 2 1 2

( ) ( ) ( ) (1 ( )) ( )
t m k m g m m k m

p S y S y S S y S   
  

      , where 
1 2

( )
m

S


  

.1 2
( ) / ( )

g m k m
y S y S


, based on Eq. (11), it can be written: 

 
.

1 2

1
( )

1 ( )
h k m

m

S
S



 


. (17) 

According to Eq. (16), it can be written: 

 
1 1 2 2

1

1 ( )
h

I II


  

  


  

,   i.e.  
1 1 2 2

1
1

I II

h

  


  
    . (17') 

Dimensionless loss I1 contains flow energy losses of the stay vane. The designing assumption 

is that terms I1 and 12 are larger than dimensionless losses in the diffuser, 2II. 

4. SUGGESTION FOR FUNCTION ( )
k

y r  

In order to reduce specific work of elementary stages near the hub, and to achieve 

flow surfaces inside the runner that do not deviate much from the cylindrical surfaces, the 

distribution function of elementary stages specific work is suggested as follows:  

 
2

0

.0 0

( ) 2 ,   for     

and   ( ) const.,         for   

k o i

k k e

y r A r B r B r r r r

y r y r r r

       


    

 (18) 

where is ( ) / 0
k

y r r    for 
0

r r . 



 Design of Small Bulb Turbines with Unequal Specific Work Distribution of the Runner's Elementary Stages  79 

Function yk (r) is defined according to radius r in 

the control cross-section 1-1 (in front of the runner).  

Function graph yk (r), which is defined by 

equation (18) is presented in Fig. 2. Function yk (r) 

is defined according to accepted parameters ro and 

yk.i = yk (ri). To obtain flow surfaces in the turbine 

runner with only a small deviation of the cylindrical 

surfaces the following is recommended: 

0 .

1
( )   and    (0.6 0.7)

2
i e k i k

r r r y Y


    , 

where 
k

Y


 is design specific work of the runner.  

The value of specific work yk.o is an unknown value and it can be defined using a 

requirement that specific work of runner (Yk) obtained by Eq. (8), is equal to the design 

specific work of runner (Yk = Yk
+
).   

Value .0ky  is determined, as will be shown later, using an iterative procedure. The 

coefficients A and B in the first Eq. (18) are calculated using formulae:  

 
2.0 .

.0 02

0

   and   A=
( )

k k i
k

i

y y
B y r B

r r


 


, (19) 

obtained from the requirement 
.

( )
k i k i

y r y , for r = ri and 0 .0( )k ky r y , for r=r0. 

Assuming that the flow surfaces in the control cross-section in front of the runner (1-

1) are cylindrical, based on Eqs. (14) and (18), it is obtained: 

 

2

1
02

1 1 0 0

,      for     

and   . ( ),    for    

z
i

z z e

c
r r r r

r r r

c const c r r r r

 
 


       

 
    

 (20) 

where the coefficients are:  

 0 2
1 3

4
h

B
r B

 

 
  

 
, 

2

1
4

h

B
B

 

 
  

 
,  

2

0

2

2
4

A r B
B




    and  0

2

4r AB



  (21) 

 Integral of the first Eq. (20), for ri  r  ro, becomes: 

 
2 2 2

1 1

1 1 1
( ) ( ( )) ( ) ( ) ln

2

o
z z o o o

o

r
c r c r r r r r

r r r
   

 
        

 
. (22) 

In the previously shown function cz1(r), for ri  r  r0 , there is an unknown velocity 

value cz1(r0) (axial component of flow velocity cz1 = cz1(r0), for r0  r  re). This velocity 

can be defined according to the requirement that the flow rate calculated by Eq. (7) is 

equal to the design turbine flow rate (Q = Q
+
). Since coefficients A and B, and also 

coefficients α, β, γ and δ, depend on unknown variable yk.0, it can be concluded that 

function cz1(r0) is determined using an iterative procedure. 

 

Fig. 2  Function yk (r) 



80 J. BOGDANOVIĆ-JOVANOVIĆ, B. BOGDANOVIĆ, I. BOŽIĆ 

Since cz1(r) = cz1(r0), for r0  r  re, and for ri  r  r0 z1c (r)  is defined by Eq. (22), Eqs. 

(7) and (8), used for determination of values yk.0 and cz1(r0), can be derived into forms: 

 
0

2 2

1 1 0 0
2 ( ) ( )( )

i

r

z z e

r

Q c r rdr c r r r      (23) 

 
0

2 2

1 .0 1 0 0

1 1
2 ( ) ( ) ( ( )( ))

i

r

k k z k z e

r

Y y r c r rdr y c r r r
Q Q

    . (24) 

5. DETERMINATION OF 
.0k

y  AND 
1 0
( )

z
c r  

Values 
.0k

y  and cz1(r0) are determined by using an iterative procedure. Perhaps, it is 

better to say that this is a double iterative procedure, since in each step of determining 

.0k
y , cz1(r0) should be determined as well by another iterative procedure [3].  

In order to reduce the number of iterative steps for determining 
.0k

y , this value is 

obtained in the first iterative step, using a formula: 

 
(1) .
.0

1

k k i
k

Y k y
y

k


 




, (25) 

where k = k (r0, ri, re)  is:   

 
2 2 2 4 4 3 3

0 0 0 0 0

2 2 2

0

( ) 0.5( ) 1.333 ( )

( )( )

i i i

e i i

r r r r r r r r
k

r r r r

    


 
. (25') 

 Formula (25) is derived using the assumption that the axial components of flow velocities in 

the control cross-section in front of the runner change negligibly (for 
1

const.
z z

c c  ). 

Using value 
(1)

.0k
y , which was determined by formula (25), the procedure of determining 

yk.0 is completed in maximum of three iterative steps. The reason for this, as calculations 

showed, is that cz1(r) values change less than specific work yk (r), for ri  r  r0. In 

iterative steps of determining yk.0 correction of this value is performed, requiring that 

calculated specific work (Yk) using formula (24) slightly differs from the design specific 

work of the turbine runner Yk
+
 ( / 0.005k k kY Y Y

 
  ). 

In each iterative procedure of determining yk.0 the value cz1(r0) is also obtained by an 

iterative procedure. In the first iterative step it can be taken that 
z1 0

c ( ) 1.2
z

r c , where: 

 
2 2

/[ ( )]
z e i

c Q r r  ,   for   Q = Q
+
. (26) 

In iterative steps of determining cz1(r0) a correction of this value is performed, and 

according to formula (23), calculated flow rate Q slightly differs from design flow rate Q
+
 

( / 0.005Q Q Q
 

  ).    

In the program for determining yk.0 and cz1(r0) [4], in the iterative procedure of solving 

the problem, cz1(r0) changes with the iterative step 0.005 zc  , and yk.0 changes with the 



 Design of Small Bulb Turbines with Unequal Specific Work Distribution of the Runner's Elementary Stages  81 

step 0.005
k

Y


  . The integrals in formulae (23) and (24) are calculated using a trapezoid 

rule, whereas in the integration area from ri to r0 calculation points are distributed for the 

iterative step Δr = 0.0025m (2.5mm). Besides printing the calculated values yk.0 and 

cz1(r0), program [4] is also developed to print values yk (r), cz1(r)  and Q(r), for r  [ri,re], 

for each iterative step Δr = 0.0025m, where r – radius in control cross-section in front of 

the turbine runner and Q(r) is the volume flow rate under the cylindrical flow surface of 

radius r. 

In the control cross-sections before (1-1) and behind (2-2) turbine runner, the flow surfaces 

are cylindrical, where 
z2

c const.
z

c  . According to Q(r) data, where r = r1(Sm), the functional 

relationship of cylinder radius in the same axisymmetric flow surfaces can be obtained. 

Regarding equation (6), for Q(Sm) = Q(r), r2(Sm) = r2(r) and z2c const. zc   it follows: 

 
2

2

( )
( )

i

z

Q r
r r r

c
 


, (27) 

where  
1

 = ( )
m

r r S and 
2 2

( )
m

r r S . 

For r2(r) < 1, as obtained for / 0ky r   , the flow surfaces in the turbine runner are 

approximately conical and r1(Sm) = r2(Sm).  

To determine the real flow surfaces' deviation from the cylindrical shape, the 

dimensionless function is 1/ 2 2( ) / ( )r r r r r . In cases where 1/ 2 ( ) 1.03r r  , it can be said that 

the real flow surface in the turbine runner negligibly vary from the cylindrical surface. 

The same calculation procedure has been used for determining values yk.0 and cz2(r0) 

and coefficients A, B, , ,  and  like in the computer program given in Ref. [4]. Even 
though the program is originally created for axial flow fans, the same code can be used for 

bulb turbine as well, replacing value h with 1/h and value cz2(r0) with cz1(r0). 

Diagrams of yk (r), cz1(r)  and 1/ 2 ( )r r  are given in Fig. 3, for yk.i = 0.7Yk
+
 and different 

values of r0 (r0 = 180, 210, 240, 280 and 310 mm) of one small bulb turbine, which is built in 

the small hydro power plant "Grcki mlin" near Prokuplje. The geometrical and design 

operating parameters of the turbine are: ri = 142 mm, re = 355 mm, Q
+ 

= 1.4m
3
/s, 

Yk
+ 

= 20.36 J/kg and   = 41.9 r/s (n = 400 min
-1

). For adopted η = 0.75 (ηm = 0,94, ηQ = 0.96 

and 1-2 = I-1 + 2-II) it follows that h.k = 0.91. As can be concluded from Fig. 3, for  

yk.i = 0.7Yk
+ 

= 14.2 J/kg, 1/ 2 ( ) 1.03r r   is obtained (when flow surfaces in the runner can be 

considered cylindrical), for r0  210 mm. It can be shown easily that for values 

yk.i = 0.6Yk
+ 

= 12.2 J/kg, 1/ 2 ( ) 1.03r r   is obtained for r0  190 mm. For the observed turbine 

it is ri / re = 0.40.  

The bulb turbine in the small hydro power plant "Grcki mlin" on the river Toplica, 

near Prokuplje, is a small turbine (PT = 26 kW); therefore, the blades are shaped using 

panel-like profiles of the runner elementary stages. The runner consists of 4 blades, and 

due to the technical reasons the blade profile thickness is: i = 16 mm near the hub and 

e = 10 mm on the blade periphery. Due to structural constraints, the profile lengths are 
li = 296 mm (ri = 142 mm) and le = 525 mm (re = 355mm). The requirement is to maintain 

geometric parameters ri = 142 mm and re = 355mm. 



82 J. BOGDANOVIĆ-JOVANOVIĆ, B. BOGDANOVIĆ, I. BOŽIĆ 

The calculation based on the model of unequal specific work of the elementary stages, 

that are changed by the rule given in equation (18), for yk.i = 0.6Yk
+ 

= 12.2 J/kg and 

r0  180 mm (when 1 0( ) 4.27 m/szc r  , yk.0 = 20.68 J/kg, A = 169.5, B = 5871,  = 47065, 

 = 104426,  = 7356,  = 408.2 and 
1/ 2
r (r) 1.023 ) the following is obtained: t.i = 31

o
 

and t.e = 15.8
o
. The inclination angle of the blade profile is t.i  t.e = 15.2

o
, which is 6.2

o
 

(29%) smaller than the blade profile designed based on the model of the equal specific 

work of the turbine elementary stages. 

The values of iterative steps and defined relative errors of specific work and volume 

flow rate can be selected in the program for determination of yk.0 and cz1(r0) [4]. However, 

the values applied in the above mentioned example of the bulb turbine give the results 

that are accurate enough for the technical practice. 

 

Fig. 3 Diagrams of ( )
k

y r , 
1
( )

z
c r  and 

1/ 2
( )r r  



 Design of Small Bulb Turbines with Unequal Specific Work Distribution of the Runner's Elementary Stages  83 

 CONCLUSION 

The distribution function of specific work in turbine elementary stages (yk(r)) are 

proposed in the paper. In addition, function  1/ 2 ( )r r  is defined, according to which the 

flow surfaces deviation compared to the cylindrical shapes can be determined.  

Using the proposed functional distribution of specific work of the turbine elementary 

stages, with the fulfilled condition that the flow surfaces negligibly deviate from the 

cylindrical surfaces, the designed bulb turbine blades are less twisted. This design method 

is applicable to stay vanes, obtaining less twisted stay vanes as well. 

Acknowledgement: This paper is result of technological project No. TR33040, Revitalization of 
existing and designing new micro and mini hydropower plants (from 100 to 1000 kW) in the 

territory of South and Southeast Serbia, supported by Ministry of Science and Technological 
Development of the Republic of Serbia.  

REFERENCES  

1. Babić, M., Stojković, S., 1990, Basics of Turbomachinery: operating principles and mathematical 

modeling, University of Kragujevac, Naučna knjiga, Beograd, Serbia (in Serbian). 

2. Benišek, M., 1998, Hydraulic Turbines, Faculty of Mechanical Engineering, Belgrade (in Serbian). 

3. Bogdanović, B., Bogdanović-Jovanović, J., Spasić, Ž., 2010, Designing of low pressure axial flow fans with 

different specific work of elementary stages, The International Conference Proceedings: Mechanical 

Engineering in XXI Century, pp. 99-102, Niš. 

4. Bogdanović, B., Bogdanović-Jovanović, J., Todorović, N., 2011, Program for determination of unequal 

specific work distribution of elementary stages in the low-pressure axial flow fan design procedure, Facta 

Universitatis, Series Mechanical Engineering, 9(2) pp.149-160. 

5. Obradović, N., 1974, Turbocompressors, Tehnička knjiga, Beograd (in Serbian). 

6. Bogdanović, B., Bogdanović-Jovanović, J., Spasić, Ž., Milanović, S., 2009, Reversible axial fan with blades 

created of slightly distorted panel profiles, , Facta Universitatis Series Mechanical Engineering, 7(1) pp.23-36 

7. Ristić B., Milenković, 1996, Small hydropower plants, Naučna knjiga, Beograd (in Serbian). 

8. Bogdanović-Jovanović, J., Bogdanović, B., Milenković, D., 2012, Determination  of  averaged  axisymmetric  

flow  surfaces  according  to  results  obtained  by   numerical  simulation  of  flow  in  turbomachinery, 

Thermal Science, 16(2) pp. 647-662. 

9. Jovičić, N., Babić, M., Jovičić, G., Gordić, D., 2005, Performance Prediction of Hydraulic Turbomachinery, 

Facta Universitatis, series Mechanical Engineering, 3(1) pp. 41-57. 

10. Albuquerque, R.B.F., Manzanares-Filho, N., Oliveira, W., 2007, Conceptual optimization of axial-flow 

hydraulic turbines with non-free vortex design, IMechE 221, Part A: J. Power and Energy, pp.713-725. 

11. Hofler, E., Gale, J., Bergant, A., 2011, Hydraulic design and analysis of the saxo-type vertical axial turbine, 

Transactions of the Canadian Society for Mechanical Engineering, 35(1), pp.119-143. 

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turbine, Renewable Energy 36, pp.2395-2403. 



84 J. BOGDANOVIĆ-JOVANOVIĆ, B. BOGDANOVIĆ, I. BOŽIĆ 

PROJEKTOVANJE MALIH CEVNIH TURBINA SA RAZLIČITIM 

JEDINIČNIM RADOVIMA ELEMENTARNIH STUPNJEVA 

OBRTNOG KOLA 

Uzimajući u obzir dosadašnja saznanja iz oblasti projetovanja turbomašina, u ovom radu je 

prikazan metod projektovanja malih cevnih turbina sa različitim jediničnim radovima elementarnih 

stupnjeva turbinskog kola u okolini glavčine. Data je funkcija raspodele jediničnih radova 

elementarnih stupnjeva turbinskog kola, pri kojoj osnosimetrične strujne površine u turbinskom kolu 

zanemarljivo malo odstupaju od cilindričnih strujnih površina. U datoj funkciji raspodele, jedinični 

rad elementarnog stupnja turbinskog kola može se, uz glavčinu, smanjiti i na 60% proračunskog 

jediničnog rada turbinskog kola, a da strujne površine u obtnom kolu budu priližno cilindrične. 

Ključne reči: cevna turbina, elementarni stupanjevi, jedinični rad, osnosimetrične strujne površine.